Generalized vector valued almost periodic and ergodic distributions

For \Cal A\subset L^1_{loc}(\Bbb J,X) let \Cal M\Cal A consist of all $f\in L^1_{loc}$ with M_h f (\cdot):=\frac {1}{h}\int_{0}^{h}f(\cdot +s)\,ds \in \Cal A for all $h>0$. Here $X$ is a Banach space, $\Bbb J= (\alpha ,\infty), [\alpha ,\infty)$ or $\Bbb R$. Usually \Cal A\subset\Cal M\Cal A\subset \Cal M^2\Cal A\subset .... The map \Cal A \to \Cal {D}'_{\Cal A} is iteration complete, that is \Cal {D}'_{\Cal {D}'_{\Cal A}}= \Cal {D}'_{\Cal A}. Under suitable assumptions \widetilde {\Cal M}^n \Cal {A}= \Cal A + \{T^{(n)} : T \in \Cal A\}, and similarly for \Cal {M}^n \Cal A. Almost periodic $X$-valued distributions \h'_{\A} with \A = almost periodic (ap) functions are characterized in several ways. Various generalizations of the Bohl-Bohr-Kadets theorem on the almost periodicity of the indefinite integral of an ap or almost automorphic function are obtained. On \Cal {D}'_{\Cal E} , \Cal E the class of ergodic functions, a mean can be constructed which gives Fourier series. Special cases of \Cal A are the Bohr ap, Stepanoff ap, almost automorphic, asymptotically ap, Eberlein weakly ap, pseudo ap and (totally) ergodic functions (\T)\E. Then always \Cal {M}^n \Cal A is strictly contained in \Cal {M}^{n+1} \Cal A. The relations between \m^n \E, \m^n\T\E and subclasses are discussed. For many of the above results a new $(\Delta)$-condition is needed, we show that it holds for most of the \A needed in applications. Also, we obtain new tauberian theorems for $f\in L^1_{loc}(\Bbb J,X)$ to belong to a class \A which are decisive in describing the asymptotic behavior of unbounded solutions of many abstract differential-integral equations. This generalizes various recent resultsComment: 69 page